Lichtenbaum-Tate duality for varieties over p-adic fields

Abstract

ABSTRACT. S. Lichtenbaum has proved in [L1] that there is a nondegenerate pairingbetween the Picard group and the Brauer group of a nonsingular projective curve C over a p -adic field K (a finite extension of the p -adic numbers Q p ). On the level of divisors the pairing is induced by the norm map Br(K ) → Br(K) for finite extensions K /K . The nondegeneracy is proven by a reduction to Tate duality for commutative group schemes over p -adic fields. This reduction is achieved by explicit cocycle calculations in Galois cohomology. The present paper introduces a new homology theory, pseudo-motivic homology, which allows us to reconstruct the above duality as a purely formal combination of a generalised form of Tate duality over p -adic fields and a form of Poincaré duality for curves over arbitrary fields due to P. Deligne. This gives a more conceptual proof of Lichtenbaum's result and an analogue in higher dimensions.

The Brauer–manin Obstruction for Zero-Cycles on Severi–brauer Fibrations Over Curves

“…Let C be a smooth projective geometrically connected curve over a field of characteristic zero, let Z be a finite nonempty collection of closed points, and let U be the complement of Z. Since Z = ?, we have 1 (see also [16]). The following lemma enables us to generalise this to open curves.…”

“…(ii) This is immediate from the above and Hilbert's Theorem 90: (1) and Z 1 Z (X/k) D are tori, and 1 H 0 (X, Z) is the total Albanese variety of X (see [16,Theorem 3.6]).…”

“…In Subsection 1.1 we will recall the definition and basic properties of pseudomotivic homology from [16]. We will also define a truncated version which will be easier to work with in later sections.…”

“…In this paper we will shed new light on the results of Frossard and remove the last cohomological condition by fitting everything in the framework of pseudo-motivic homology. This is a homology theory defined in [16] by 1 H i (X, Z) := Ext −i K sm (RΓ(X/K, G m ), G m ). For i = 0 this gives a homology group that is in between the group of K-points of the total Albanese variety and the Chow group of zero-cycles.…”

Об Элементарном Препятствии К Существованию Рациональных Точек

“…Однако и здесь (как и в более общем случае поля частных гензелева кольца дискретного нормирования с конечным полем вычетов) из зануления e(X) следует зануление δ(X) (см. [12], а также [4]). Однако для некоторых других полей, например, для поля рядов Лорана C((t)) это уже неверно: как показал недавно О. Виттенберг, e(X) = 0 для любого многообразия над полем когомологической размерности 1.…”